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Properties

Label 28.0.792...344.1
Degree $28$
Signature $[0, 14]$
Discriminant $7.925\times 10^{50}$
Root discriminant $65.74$
Ramified primes $2, 43$
Class number $43$ (GRH)
Class group $[43]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 3*x^26 + 18*x^24 + 304*x^22 - 431*x^20 + 627*x^18 + 12915*x^16 - 6007*x^14 + 76379*x^12 - 24573*x^10 + 152173*x^8 - 11756*x^6 + 110547*x^4 - 8906*x^2 + 6241)
 
gp: K = bnfinit(x^28 - 3*x^26 + 18*x^24 + 304*x^22 - 431*x^20 + 627*x^18 + 12915*x^16 - 6007*x^14 + 76379*x^12 - 24573*x^10 + 152173*x^8 - 11756*x^6 + 110547*x^4 - 8906*x^2 + 6241, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6241, 0, -8906, 0, 110547, 0, -11756, 0, 152173, 0, -24573, 0, 76379, 0, -6007, 0, 12915, 0, 627, 0, -431, 0, 304, 0, 18, 0, -3, 0, 1]);
 

\( x^{28} - 3 x^{26} + 18 x^{24} + 304 x^{22} - 431 x^{20} + 627 x^{18} + 12915 x^{16} - 6007 x^{14} + 76379 x^{12} - 24573 x^{10} + 152173 x^{8} - 11756 x^{6} + 110547 x^{4} - 8906 x^{2} + 6241 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(792537323068373529244880273632877655015215903801344\)\(\medspace = 2^{28}\cdot 43^{26}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $65.74$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 43$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $28$
This field is Galois and abelian over $\Q$.
Conductor:  \(172=2^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{172}(1,·)$, $\chi_{172}(131,·)$, $\chi_{172}(97,·)$, $\chi_{172}(133,·)$, $\chi_{172}(65,·)$, $\chi_{172}(137,·)$, $\chi_{172}(11,·)$, $\chi_{172}(145,·)$, $\chi_{172}(75,·)$, $\chi_{172}(107,·)$, $\chi_{172}(21,·)$, $\chi_{172}(151,·)$, $\chi_{172}(127,·)$, $\chi_{172}(27,·)$, $\chi_{172}(161,·)$, $\chi_{172}(35,·)$, $\chi_{172}(39,·)$, $\chi_{172}(41,·)$, $\chi_{172}(171,·)$, $\chi_{172}(45,·)$, $\chi_{172}(47,·)$, $\chi_{172}(113,·)$, $\chi_{172}(51,·)$, $\chi_{172}(121,·)$, $\chi_{172}(87,·)$, $\chi_{172}(59,·)$, $\chi_{172}(125,·)$, $\chi_{172}(85,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{79} a^{23} - \frac{26}{79} a^{21} - \frac{10}{79} a^{19} - \frac{17}{79} a^{17} - \frac{21}{79} a^{15} - \frac{19}{79} a^{13} + \frac{33}{79} a^{11} - \frac{7}{79} a^{9} + \frac{29}{79} a^{7} - \frac{2}{79} a^{5} + \frac{2}{79} a^{3} + \frac{36}{79} a$, $\frac{1}{59824093} a^{24} + \frac{408957}{59824093} a^{22} + \frac{1439751}{8546299} a^{20} + \frac{1682122}{8546299} a^{18} - \frac{6127419}{59824093} a^{16} - \frac{23873345}{59824093} a^{14} - \frac{27757328}{59824093} a^{12} + \frac{14546737}{59824093} a^{10} + \frac{16919459}{59824093} a^{8} + \frac{25292559}{59824093} a^{6} - \frac{15757338}{59824093} a^{4} - \frac{28138816}{59824093} a^{2} + \frac{32748}{757267}$, $\frac{1}{59824093} a^{25} - \frac{348310}{59824093} a^{23} + \frac{4252457}{8546299} a^{21} + \frac{2763932}{8546299} a^{19} + \frac{6746120}{59824093} a^{17} - \frac{7970738}{59824093} a^{15} - \frac{13369255}{59824093} a^{13} - \frac{10443074}{59824093} a^{11} + \frac{22220328}{59824093} a^{9} + \frac{3331816}{59824093} a^{7} - \frac{14242804}{59824093} a^{5} - \frac{29653350}{59824093} a^{3} - \frac{24674520}{59824093} a$, $\frac{1}{64484380879649266112000958051623} a^{26} - \frac{1846688766049374684735}{256909883982666398852593458373} a^{24} + \frac{8116597678040179047644682410103}{64484380879649266112000958051623} a^{22} + \frac{1835577246686412892805105628562}{9212054411378466587428708293089} a^{20} - \frac{7521040500140867073950144623219}{64484380879649266112000958051623} a^{18} - \frac{15620768505729586407761717743904}{64484380879649266112000958051623} a^{16} - \frac{27329270793042973961807632781662}{64484380879649266112000958051623} a^{14} + \frac{24844741773899302045492252878262}{64484380879649266112000958051623} a^{12} + \frac{11988504890918784700348330870079}{64484380879649266112000958051623} a^{10} + \frac{8157749178961277644332202332870}{64484380879649266112000958051623} a^{8} - \frac{8437831942498158985693015813948}{64484380879649266112000958051623} a^{6} - \frac{29553881544262952442842734858807}{64484380879649266112000958051623} a^{4} - \frac{23116206775666946277684895206984}{64484380879649266112000958051623} a^{2} + \frac{2225892109010122591415257427}{10332379567320824565294176903}$, $\frac{1}{64484380879649266112000958051623} a^{27} - \frac{1846688766049374684735}{256909883982666398852593458373} a^{25} - \frac{45982180143272358937717343267}{64484380879649266112000958051623} a^{23} + \frac{4517567771518118354967894118955}{9212054411378466587428708293089} a^{21} + \frac{9620377202044380879872894858858}{64484380879649266112000958051623} a^{19} - \frac{5825672675909444719862838039860}{64484380879649266112000958051623} a^{17} + \frac{15116144469510973352420845935862}{64484380879649266112000958051623} a^{15} - \frac{13519383559562919565445025962577}{64484380879649266112000958051623} a^{13} + \frac{560893089461952731132971215361}{64484380879649266112000958051623} a^{11} + \frac{811427306596171378408042554837}{64484380879649266112000958051623} a^{9} + \frac{12784875688778814671421223544814}{64484380879649266112000958051623} a^{7} - \frac{13228721827896049629677935352067}{64484380879649266112000958051623} a^{5} + \frac{25043014387615417021151263337899}{64484380879649266112000958051623} a^{3} - \frac{278779224350316596151138446999}{816257985818345140658239975337} a$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

$C_{43}$, which has order $43$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{34746942274768009}{733138626249524197283} a^{27} - \frac{90849002669170407}{733138626249524197283} a^{25} + \frac{604982509902124085}{733138626249524197283} a^{23} + \frac{1533783770301570770}{104734089464217742469} a^{21} - \frac{10551754272131332116}{733138626249524197283} a^{19} + \frac{21953131397543139958}{733138626249524197283} a^{17} + \frac{445553048340535517168}{733138626249524197283} a^{15} - \frac{27840992264299803552}{733138626249524197283} a^{13} + \frac{2840242315285543325081}{733138626249524197283} a^{11} - \frac{78138797732291591290}{733138626249524197283} a^{9} + \frac{6149681341485093545410}{733138626249524197283} a^{7} + \frac{917775515316363541979}{733138626249524197283} a^{5} + \frac{4884488566730410537874}{733138626249524197283} a^{3} + \frac{10450539251070167142}{9280235775310432877} a \) (order $4$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 2714544445552.786 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 2714544445552.786 \cdot 43}{4\sqrt{792537323068373529244880273632877655015215903801344}}\approx 0.154922221619408$ (assuming GRH)

Galois group

$C_2\times C_{14}$ (as 28T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{43}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-43}) \), \(\Q(i, \sqrt{43})\), 7.7.6321363049.1, 14.14.28152039412241052225421312.1, 14.0.654698590982350051753984.1, 14.0.1718264124282290785243.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{4}$ R ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
43Data not computed